Stochastic Model Predictive Control for
Constrained Networked Control Systems
with Random Time Delay
Panagiotis Patrinos∗ Pantelis Sopasakis∗∗ Haralambos Sarimveis∗∗
∗ Department of Mechanical and Structural Engineering, University of Trento, Via Mesiano 77, 38100, Italy. (e-mail: [email protected]).
∗∗National Technical University of Athens, Heroon Politechniou 9, NTUA Zografou Campus, GR-15780 , School of Chemical Engineering,
Unit of Automatic Control & Informatics, (e-mail: chvng,[email protected])
Abstract: In this paper the continuous time stochastic constrained optimal control problem is formulated for the class of networked control systems assuming that time delays follow a discrete-time, finite Markov chain . Polytopic overapproximations of the system’s trajectories are employed to produce a polyhedral inner approximation of the non-convex constraint set resulting from imposing the constraints in continuous time. The problem is cast in a Markov jump linear systems (MJLS) framework and a stochastic MPC controller is calculated explicitly, offline, coupling dynamic programming with parametric piecewise quadratic (PWQ) optimization. The calculated control law leads to stochastic stability of the closed loop system, in the mean square sense and respects the state and input constraints in continuous time.
1. INTRODUCTION
A networked control system (NCS) is a feedback control system where its various components (sensor, actuator, controller, plant) are connected through a communication network. Due to their numerous advantages such as low installation and maintenance cost and high flexibility, they have attracted a lot of interest over the past few years (Antsaklis and Baillieul [2004], Baillieul and Antsaklis [2007], Heemels and van de Wouw [2010], Hespanha et al. [2007], Tipsuwan and Chow [2003], Zhang et al. [2001]). The presence of a communication network in the loop introduces many interesting phenomena such as random transmission delays and packet losses, which need to be taken into account by the control synthesis algorithm in order to guarantee closed-loop stability. For example the presence of time-varying transmission delays may render, an otherwise stable closed-loop system, unstable (Cloosterman [2008], Zhang et al. [2001]).
There are two approaches proposed in the literature con-cerning the modeling of the transmission delay. The one is to model it as a deterministic quantity that takes values in a closed interval (deterministic approach), (Clooster-man [2008], Clooster(Clooster-man et al. [2006], Clooster(Clooster-man et al. [2008], Cloosterman et al. [2009]). The second approach is to consider the delay as random (stochastic approach), (Antunes et al. [2009], Chen et al. [2008a], Chen et al. [2008b], Donkers et al. [2010], Montestruque and Antsaklis [2004], Nilsson [1998], Nilsson et al. [1998], Seiler and Sengupta [2005], Shi and Yu [2009], Zhang et al. [2005]). The first approach leads to an uncertain discrete-time system, with the uncertainty appearing in an exponential form. The next step of the deterministic approach is to
construct a polytopic (possibly with an additive norm-bounded term) over-approximation of the exponential un-certainty, in order to arrive to a model that is amenable to robust stability analysis and control synthesis, using classical LMI (linear matrix inequalities) techniques. The stochastic approach uses a probabilistic description of the transmission delay, assuming that it is either an indepen-dent across time random variable that follows a continuous or discrete probability distribution, or a stochastic process such as a Markov chain. Therefore, stability analysis and control synthesis is performed in a stochastic setting with the appropriate notion of stability being that of mean-square stability.
On the other hand, constraints on the input and state of the system usually need to be imposed in any realistic setting. Usually constrained control problems are treated in a model predictive control (MPC) framework, by solving a finite-horizon constrained optimal control problem at each sampling instant. Assuming that the control action is imposed through zero-order hold (ZOH) to the system, in-corporation of input constraints at the sampling instances to the MPC optimization problem is straightforward. How-ever, since the actual system is continuous in time, the requirement that the state constraints must be satisfied at all times results in a non-convex optimization problem even for linear systems with polyhedral constraints. To the best of our knowledge, there are no practical approaches considered in the literature concerning constrained con-trol of (even non-networked) sampled-data systems, where state constraints are imposed in the continuous time. In this paper, we consider control of constrained, continuous-time linear systems through a communication network. We
model the transmission delay from the sensor to controller as a Markov chain, while we assume that the delay from the controller to the actuator (including the controller delay itself) is constant by a buffering technique. Specifi-cally, we assume that the controller has knowledge of the current sensor-to-controller delay at each time instant (as is usually the case via timestamping techniques), and that the value of the delay switches among a finite number of values according to a Markov chain. The system is subject to hard state and input constraints that must be fulfilled in continuous time by the closed-loop system. The model is cast in a Markovian switching framework, which is analyzed in the authors recent work, Patrinos and Sarimveis [2010]. The state constraints in the continuous time are approximated by imposing sufficient conditions that the state of the system must satisfy only at the sampling instances. The technique is based on recent ideas regarding polytopic approximations of systems with time-varying delays, (Cloosterman [2008], Gielen et al. [2009], Gielen et al. [2010], Heemels et al. [2010], Hetel et al. [2006], Hetel et al. [2007]).
Advantages of the proposed approach is that it guarantees fulfillment of constraints for the continuous-time system for all possible values of the transmission delay, without destroying the structure of the resulting optimal control problem, i.e. this is guaranteed by imposing the state to lie in a polyhedral set only at the sampling instances. Fur-thermore, the controller is allowed to switch synchronously with the sensor-to-controller delay providing larger flexi-bility and less conservativeness in comparison to the de-terministic approach.
2. MODEL DEFINITION 2.1 NCS model
The NCS model consists of a linear, time-invariant, continuous-time plant and a discrete-time controller that are connected through a communication network with induced sensor-to-controller (SC), τsc and controller-to-actuator (CA), τca, delays. The controller delay (the time needed by the controller to perform computations) is as-sumed to be incorporated into the CA delay. The full state of the system is sampled by a time-driven sensor with a constant sampling interval h > 0. The discrete-time controller is event-driven and able to monitor the SC delay, via timestamping. The CA delay is considered to be constant by using the buffering technique. The discrete-time control signal uk is transformed to a continuous-time control input u(t) by a zero-order hold device (ZOH). Based on these assumptions, the NCS model is:
˙ x(t) = Acx(t) + Bcu(t) (1a) u(t) = uk, for t ∈ [kh + τksc+ τ ca k , (k + 1)h + τ sc k+1+ τ ca k+1) (1b) with Ac ∈ Rn×n, Bc ∈ Rn×m. The SC delay evolves according to a discrete-time, time-homogeneous Markov chain {rk}k∈N taking values in a finite set S , {1, . . . , S} with transition matrix P = (pij) ∈ RS×S. Therefore, τksc switches among a finite set of values, where the switching is orchestrated by the underlying Markov chain {rk}k∈N. This assumption manages, in some sense, the correlation of the transmission delays across time that is present in many
types of networks (Nilsson [1998], Tipsuwan and Chow [2003]). Since the CA delay is considered constant through the buffering technique, the two transmission delays can be lumped into τk, τksc+ τkca. Therefore, the evolution of the total transmission delay τk is governed by the underlying Markov chain {rk}k∈N and the controller has access to τk at time k, but only probabilistic information is available regarding its future evolution. Throughout the paper, it is assumed that τk ≤ h. Furthermore, there is one-to-one correspondence between the possible values of τk and the states of the Markov chain, i.e.:
τk= τi , τsc,i+ τca, if rk= i, i ∈ S (2) The cover of a state i ∈ S is the set of all states accessible i in one time step, i.e. Si, {j ∈ S|pij > 0}. An admissible switching path of length N ∈ N, r , (r0, · · · , rN) for {rk}k∈N is a switching path for which rk+1∈ Srk, for any
k ∈ N[0,N −1]. We denote by G the set of all admissible switching paths (of infinite length), and by GN the set of all admissible switching paths of length N . For any i ∈ S, G(i) , {r ∈ G|r0 = i} and GN(i) , {r ∈ GN|r0 = i} denote the set of all admissible switching paths emanating from i , of infinite length and length N , respectively. The state of the system generated by the NCS model (1) is required to belong to a polyhedral set X, i.e.
x(t) ∈ X, t ∈ R+ (3)
Notice that constraints are imposed in the continuous time. Similarly, the input signal is required to belong to a polyhedral set U , i.e.
uk∈ U, k ∈ N (4)
Notice that since the input is a discrete-time signal, imposing the constraints only at the sampling instances is sufficient to guarantee satisfaction of input constraints in continuous time. We are interested in mode-dependent policies of the form π = {µ0, µ1, . . .}, with µk : Rn × S → Rm. Given a policiy π and a switching delay path r, the input signal in continuous-time is given by u(t) = µk(x(kh), rk), for t ∈ [kh + τk, (k + 1)h + τk+1). Given a policy π, an initial state x(0) = x, an initial delay r0= i, and an admissible switching path r, the solution of (1) at time t ∈ R+ is denoted by ψ(t; x, i, π, r).
2.2 Exact discrete-time system The exact discretization of (1) is xk+1= eAchxk+ Z h−τk 0 eAcsdsB cuk+ Z h h−τk eAcsdsB cuk−1
where xk , x(kh) is the discrete-time state at the kth sam-pling instant. The above system can be cast in standard state-space form with respect to the augmented state vec-tor ξk , [x0k u0k−1]0, (Cloosterman [2008], Nilsson [1998]).
ξk+1= Arkξk+ Brkuk (5) with Ai , h eAchRh h−τ ie AcsdsB c 0 0 i , Bi , Rh−τ i 0 e AcsdsB c I , i ∈ S. Notice that (Ai, Bi), i ∈ S can be computed ex-actly. Therefore, the NCS model (1) has been transformed into the Markovian Jump Linear system (MJLS) (5) with respect to the augmented state vector ξk, via exact dis-cretization.
2.3 Handling constraints in continuous time
Guaranteeing constraint fulfillment for the closed-loop sys-tem requires from the controller to take into consideration the so-called inter-sample behavior, i.e. what happens to the state vector between two sampling instants. Let ˜t , t− kh. Assume that rk = i. Then the inter-sample behavior is described by the following two equations (Cloosterman [2008]): x(kh + ˜t) = Γ1i(˜t)ξk, for ˜t ∈ [0, τi) (6a) x(kh + ˜t) = Γ2i(˜t)ξk uk , for ˜t ∈ [τ i, h) (6b) where Γ1i(˜t) ,heAc ˜tR˜t 0e AcsdsB c i (7a) Γ2i(˜t) ,heAc ˜tR˜t ˜ t−τ ie AcsdsB c Rt−τ i˜ 0 e AcsdsB c i (7b) Considering ˜t as an uncertain parameter appearing in the matrices Γji(˜t), j = 1, 2 in exponential form, one can compute an over-approximation separately for these two cases. Specifically, let T1
i , [0, τi), Ti2 , [τi, h). Furthermore, let: Γji , {Γji(˜t)|˜t ∈ Tij} , j = 1, 2 (8a) ˜ Γji , Pij X p=1 αpΓji,p α ∈ RP j i + , Pij X p=1 αp= 1 , j = 1, 2 (8b) where Γ1
i,p ∈ Rn×(n+m), Γ2i,p∈ Rn×(n+2m) and P j i, i ∈ S, j = 1, 2 are positive integers. The next lemma provides a sufficient condition imposed on the the state-input vector of the exact discretization (5) such that the state constraints (3) of the NCS (1) are satisfied in continuous time. Its proof is omitted due to lack of space.
Lemma 1. Let rk = i and Γ j i, ˜Γ j i be as in (8) with Γji ⊆ ˜Γji, j = 1, 2. Let ˜Xi , (Xi1 × Rm) ∩ Xi2, where Xij,T p∈Pij(Γ j i,p)−1(X), j = 1, 2. Then: (ξk, uk) ∈ ˜Xi⇒ x(t) ∈ X, ∀ t ∈ [kh, (k + 1)h) (9) Notice that ˜Xi is a polyhedral set. For every i ∈ S, let:
Yi, ˜Xi∪ (Rn× U ) (10) Then, it follows from lemma 1 that the state and input constraints (3), (4) for the continuous time system are satisfied for all possible delay switching paths if:
(ξk, uk) ∈ Yrk, k ∈ N ∀ r ∈ G (11)
Summing up, the polyhedral state constraint set of the continuous-time NCS (1), is inner-approximated with a family of mode-dependent polyhedral sets that capture the dependence on the random time delay τk , with respect to the state vector and input vector of the exact discretization (5).
2.4 Polytopic Overapproximation of the trajectories of an NCS
In this section we will present a method for the overapprox-imation of Γji , by polytopes of matrices, ˜Γji . Taking into account the special structure of Γji(t), this boils down to overapproximating functions of the general form ∆0(t) , eActfor t ∈ [t
1, t2] and Γ0(t) , Rt
0e AcsdsBc, i.e., we have
to determine sets C∆0 and CΓ0 such that for all t ∈ [t1, t2],
{∆0(t)|t ∈ [t1, t2]} ⊆ C∆0 and {Γ0(t)|t ∈ [t1, t2]} ⊆ CΓ0.
Heemels et al. [2010] provide a thorough literature overview on that matter from which the approach based on the canonical Jordan form decomposition and the Cayley-Hamilton Theorem provide a framework for the calculation of such polytopic overapproximations. Other approaches that employ truncated Taylor series expansions lead to non-polytopic norm bounded overapproximations. Finally, the method of Gridding and Bounding (G&B) Heemels et al. [2010] yields tight overapproximations and can be employed to obtain arbitrarily tight -overapproximations by partitioning the interval [t1, t2] using a sequence of points t1 = τ0 < τ1 < · · · < τs = t2 and applying the overapproximation procedure on each subinterval [tl, tl+1], l ∈ N[0,s−1].
Applying Jordan decomposition on Ac , we obtain the equivalent representation Ac = QJ Q−1 , hence ∆0(t) and Γ0(t) are rewritten as ∆0(t) = Q eJ tQ−1, Γ0(t) = QRt
0e
J zdzQ−1B. The special structure of the Jordan matrix J allows for the analytical calculation of eJ tand its integral Rt
0e
J zdz and via simple algebraic manipulations upper and lower bounds can be calculated element-wise. In particular, ∆0(t) and Γ0(t) are decomposed to ∆0(t) = Pκ
i=0ϕi(t)Si and Γ0(t) = P κ
i=0γi(t)SiB, where Si are sparse matrices with 0−1 entries. Exploiting the analytical formulas of ϕi(t) and γi(t) we calculate upper and lower bounds for them on [t1, t2], that is:
γ
i≤ γi(t) ≤ γi and ϕi≤ ϕi(t) ≤ ϕi (12) This leads to the following overapproximations:
{∆0(t)|t ∈ [t1, t2]} ⊆ C∆0 , conv{Fj}j∈N[1,2κ ] =nPκ i=1µiSi|µi∈ {ϕi, ϕ} o {Γ0(t)|t ∈ [t1, t2]} ⊆ CΓ0 , conv{Gj}j∈N[1,2κ ] =nPκ i=1µiSiB|µi∈ {γi, γ} o
However, this approach introduces a constant overap-proximation error. To surmount this drawback, we parti-tion equidistantly the interval [t1, t2] into s subintervals Ti , [τl, τl+1] thus obtaining sets C∆0,l and CΓ0,l,l ∈
N[0,s−1]. Then the overall overapproximations are CΓs0 ,
conv{CΓ0,l}l∈N[0,s−1] and C
s
∆0 , conv{C∆0,l}l∈N[0,s−1].
3. FROM CONTINUOUS-TIME TO DISCRETE-TIME STOCHASTIC OPTIMAL CONTROL
In this section we formulate the constrained optimal con-trol problem for the NCS as a stochastic concon-trol problem in continuous time. However, since the input to the system is a signal in discrete time, the expected outcome from the solution to the optimization problem is an optimal control policy π? , {µ?0, µ?1, . . .}. Given an initial state x(0) = x, an initial mode r0 = i and a policy π , {µ0, µ1, . . .}, we introduce the infinite horizon cost function:
Vπ(x, i) , E Z ∞ 0 g(x(t), u(t))dt (13) where g is a convex quadratic function of u, i.e., g(x, u) ,
1 2(x
positive definite respectively), x(t) , ψ(t; x, i, π, r) and u(t) , µk(xk, rk) for t ∈ [kh+τk, (k+1)h+τk+1) (piecewise constant using a ZOH element). The constrained infinite horizon stochastic control problem for the NCS is then formulated as follows:
V?(x, i) , inf{Vπ(x, i)|π ∈ Π(x, i)} (14) where P i(x, i) is the set of admissible policies for initial state x and initial mode i, i.e.:
Π(x, i) , π ψ(t; x, i, π, r) ∈ X, t ∈ R+ µk(ψ(t; x, i, π, r), rk) ∈ U, k ∈ N, ∀r ∈ G(i)
The continuous-time stochastic optimal control prob-lem (14) can be transformed into a discrete-time one ex-ploiting the exact discretization of the system. The infinite horizon cost function (13) is restated as:
Vπ(x, i) = E "∞ X k=0 1 2 Z (k+1)h kh (x(t)0Qx(t) + u(t)0Ru(t))dt #
Some simple algrebraic manipulations yield: Vπ(ξ, i) = E "∞ X k=0 `(ξk, uk, rk) # (15) where `(ξ, u, i) , 12[ ξ u] 0hQ i Si S0 i Ri i [ξ u] (16)
The formulas for the matrices appearing in (16) are omitted due to lack of space.
4. FINITE-HORIZON CONSTRAINED STOCHASTIC OPTIMAL CONTROL FOR NCS
The continuous-time NCS (1) subject to constraints (3) and (4) has been transformed to the Markovian switch-ing system (5) subject to the mode dependent polyhe-dral constraint sets (11). Notice that the discrete-time system˜eqrefeq:MJLS has resulted from exact discretiza-tion, and approximation is performed with respect to the continuous-time state constraints only. Therefore, the ob-jective has now become to design an optimal controller (in some sense) for the MJLS (5). This problem is a spe-cial case of the generic framework introduced in Patrinos and Sarimveis [2010]. Specifically, since individual mode dynamics are linear and the mode-dependent constraint sets are polyhedral, the MPC problem can be solved ex-plicitly by coupling dynamic programming with the con-vex parametric piecewise quadratic solver of Patrinos and Sarimveis [2011].
For each i ∈ S, let Ui(ξ) , {u ∈ Rm|(ξ, u) ∈ Yi} and Ξi , dom Ui. Let Y , {Yi}i∈S and Ξ , {Ξi}i∈S. A mapping µ : Rn+m× S → Rm, such that µ(ξ, i) ∈ U
i(ξ) for each ξ ∈ Ξi, i ∈ S, is called a (mode-dependent) control law. An infinite sequence of control laws π = {µ0, µ1, . . .} is called a policy. If the policy is of the form {µ, µ, . . .} then it is called stationary and is denoted by µ. Since we are only dealing with mode-dependent control laws and policies, the adjective mode-dependent will be omitted for brevity for the rest of the paper.
The solution of (5) at time k given a policy π and a switching path r with r0 = i and ξ0 = ξ is denoted by φ(k; ξ, i, π, r). For ξ ∈ Ξi, i ∈ S, we denote by ΠN(ξ, i) the set of admissible policies of length N , i.e.
ΠN(ξ, i) , π ξk= φ(k; ξ, i, π, r), k ∈ N[0,N ] (ξk, µk(ξk, rk)) ∈ Yrk, k ∈ N[0,N −1] ξN ∈ ΞfrN∀r ∈ GN(i)
Here Ξf , {Ξfi}i∈S is the mode-dependent terminal set, to which the state of the system needs to be steered at the end of the horizon. The finite horizon cost of the admissible policy π ∈ ΠN(ξ, i) for (5) is:
VN,π(ξ, i) , E "N −1 X k=0 `(ξk, uk, rk) + Vf(ξN, rN) # (17) where ξk, φ(k; ξ, i, π, r), uk , µk(ξk, rk) and the terminal cost Vf(ξ, i) , ξ0Pifξ with P
f
i positive semidefinite, i ∈ S. The constrained finite horizon stochastic optimal control problem is:
PN(ξ, i) : VN?(ξ, i) , inf π∈ΠN(ξ,i)
VN,π(ξ, i) (18) We call VN? the finite horizon value function. Notice that optimization is sought over truly closed-loop policies. Tak-ing into account lemma 1, this guarantees satisfaction of the constraints for the continuous-time system for every admissible transmission delay path. The multi-stage prob-lem PN(ξ, i) can be decomposed in one-stage problems and solved using dynamic programming (DP). The dynamic programming algorithm for (18) is:
V0?= Vf (19a) Vk+1? (ξ, i) = inf u {`(ξ, u, i) + X j∈S pijVk?(Aiξ + Biu, j)}, i ∈ S, k ∈ N[0,N −1] (19b) Since the individual mode dynamics of the Markovian switching system (5) are linear, the mode-dependent con-straints (11) are polyhedral, and the stage cost (16) and terminal cost Vf are convex quadratic, it follows from theorem VI.I of Patrinos and Sarimveis [2010], that V?
k(·, i) are convex PWQ for each i ∈ S and each k ∈ N[1,N ]. Furthermore, the DP subproblems can be explicitly solved off-line by the convex parametric PWQ optimization al-gorithm of Patrinos and Sarimveis [2011]. The alal-gorithm decomposes the state space into a finite number of critical regions in a graph traversal framework, using graphical derivative (Rockafellar and Wets [2009]) formulas of so-lution mappings. For each k ∈ N[0,N −1] , the proposed algorithm calculates the optimal mode-dependent control law µ?
kas a piecewise affine (PWA) mapping for each mode i ∈ S, i.e. dom Vk?(·, i) is decomposed in a finite number of polyhedral sets {Rjk,i}j∈Jk,i (Jk,i is a finite index set)
on each of which µ?k is affine, i.e. µ?k(ξ, i) = Kk,ij ξ + κjk,i, if ξ ∈ Rjk,i.
5. STOCHASTIC MPC OF CONSTRAINED NCS Since PN(ξ, i) is solved explicitly using dynamic program-ming and the convex parametric PWQ solver, at the last step of the algorithm we obtain the explicit stochastic MPC (SMPC ) law, i.e. the control law that would result by solving repetitively online the problem PN(ξk, rk) at each time instant k, and applying to the system µ?
N(ξk, rk). In fact the procedure could be performed on-line by refor-mulating PN(ξk, rk) as a multi-stage stochastic optimiza-tion problem (see secoptimiza-tion VI.B of Patrinos and Sarimveis
[2010]) using a scenario-based approach. The resulting optimization problem is a large-scale quadratic program, with an exponential increase in complexity with respect to the prediction horizon. Instead, the explicit DP approach decomposes the multi-stage problem into smaller single-stage problems, and performs all the heavy computations off-line. Therefore, online, only the evaluation of a PWA affine mapping is needed in order to compute the MPC law. For future reference, the following notation for the Markovian switching system in closed-loop with the SMPC controller is introduced:
ξk+1= Arkξk+ Brkµ
?
N(ξk, rk) (20) 6. STABILITY AND INVARIANCE PROPERTIES OF
STOCHASTIC MPC FOR NCS
In this section we provide conditions that the terminal cost and terminal set must satisfy in order for the closed-loop system (20) with SMPC to be mean-square (MS) stable. Paralleling the theory of deterministic MPC for linear systems, the terminal weighting matrices Pif can be se-lected as the solution of the infinite horizon unconstrained stochastic optimal control problem, i.e. as the solution of the CARE Costa et al. [2005], ch.4):
Pif = A0iEi(Pf)Ai − (A0iEi(Pf)Bi+ Si)· (21) (Ri+ Bi0Ei(Pf)Bi)−1(B0iEi(Pf)Ai+ Si) + Qi, i ∈ S where Ei(Pf) , Pj∈SpijPif, and Ξ
f = {Ξf i}i∈S is the maximal uniformly positive invariant set for the Markovian switching system in closed loop with the unconstrained optimal policy µ(ξ, i) , −(Ri + B0iEi(Pf)Bi)−1(Bi0Ei(Pf)Ai+Si)ξ (Patrinos and Sarimveis [2010]). The following theorem provides the promised sta-bilizing properties of the SMPC controller. Its proof can be found in Patrinos and Sarimveis [2010].
Theorem 2. Suppose that the terminal cost and terminal set are selected as above for the Markovian switching system (5) subject to (11). Then the origin is MS stable in Ξ?
N , domVN?for the system in closed loop with the SMPC controller (cf. (20)). Furthermore, if 0 ∈ int Ξfi, i ∈ S then the origin is exponentially MS stable in Ξ?N.
Theorem 2 in turn implies MS stability of the origin for the continuous time NCS (1) in closed-loop with the SMPC controller, since using lemma 4.3.5 of Cloosterman [2008], that provides an upper bound for the norm of the state of the continuous time system between two sampling instants of the form ||x(kh + r)|| ≤ c0||xk|| + c1||uk|| + c2||uk−1||, one concludes that limt→∞E[x(t)]2= 0 .
7. EXAMPLE
In this section we apply the proposed method on an applied control problem and manifest its advantages over alternative approaches found in literature (Cloosterman [2008]). The NCS describes a printer that is controlled through a network. The matrices in (1) are Ac = [0 10 0], Bc= [126.700 ], while X = [−10, 10]
2, U = [−2, 2], Q = 10I 2 and R = 1. The sampling interval is h = 20 ms while the SC delay can take the values τsc,1= 3 ms and τsc,2= 15 ms with transition matrix P = [0.67 0.33
0.30 0.70]. The CA delay is considered constant with τca= 1 ms. We set the prediction
horizon to N = 10 steps. In the following illustrations we present a visualization of the polyhedral decomposition of the feasible state space on which the control law is defined as a PWA function over these regions. The mode-dependent PWA control law consists of 61 and 73 critical regions (cf. Figure 1) for each of the two modes.
In order to elucidate the benefits of SMPC we compare our results with alternative control approaches. The first ap-proach (Delay-free MPC ) is a deterministic MPC scheme for the exact discretization of the continuous-time system without taking into consideration the time-varying delay i.e., for the system xk+1 = eAchxk+ Γ0(h). Constraints are imposed only on discrete sampling times while the cost function is considered to be quadratic, `(x, u) =
1 2(x
0Q
hx + u0Rhu) where Qh = hQ and Rh = hR. The second alternative scheme (Non-switched MPC ) is a de-terministic MPC controller for the exact discretization of the continuous-time system where the delay is considered constant and equal to its greatest value (worst case sce-nario, τmax = 16 ms), i.e. for the discrete-time system ξk+1 = [eAch Γ0(h)−Γ0(h−τmax)] ξk + Γ0(h − τmax)uk and the constraints are imposed only for the sampling times. In order to compare SMPC against the alternative schemes, 20 simulations (corresponding to 20 switching paths according to the transition matrix) for every ex-treme point of the effective domain of V?
N(·, i), i ∈ S are performed. For every single one of them, SMPC achieved stochastic stability in the mean-square sense for the continuous time closed loop system while respecting the constraints in the continuous time. Non-switched MPC achieved this goal only in 66.77% of the cases while for delay-free MPC the percentage drops to 8.47%. An il-lustrative simulation of the NCS in closed-loop with the SMPC controller is depicted in Figure 2.
Fig. 1. PWA control law over a polyhedral decomposition of the extended state space for the network controlled printer.
ACKNOWLEDGEMENTS
The work of the first author was financially supported by the National Scholarship Foundation of Greece and the NTUA Senator Committee of Basic Research Program 65/1631. The work of the second author was financially supported by the NTUA Senator Committee of Basic Research Program 65/1867.
Fig. 2. Simulation of the closed-loop system using the SMPC controller, in continuous time, starting from x(0) = [9.72 8.98]0 and r0= 1.
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